The proof we give is based on the work of Heap, Radziwiłł and Soundararajan [1]} which in turn is based on the method introduced in Radziwiłł and Soundararajan [2]}.
The general principle in these works is that if one can compute the \(2k^\text{th}\)  moment of a given \(L\) -function twisted by an arbitrary Dirichlet polynomial then one can find upper bounds of the right order for all of its lower order moments.
In particular, this approach is used to prove Theorem REF  in the case \(h = 0\) .
We combine the ideas of the paper [1]} with twisted joint moment calculations to deduce Theorem REF  in the case of \(h = 1\)  and then deduce the result from Hölder's inequality– the bounds we obtain are of the right order since the exponent of \(\log T\)  in (REF ) is linear in \(h\) .
We are forced to take \(k\in [1,2]\)  because \(2k - 2h\)  is only nonnegative when \(k \ge 1\)  at the boundary case \(h = 1\) .
It is likely that one could establish sharp bounds on \(I_T(k,h)\)  and \(J_T(k,h)\)  in the full range \(k > -\tfrac{1}{2}\)  and \(-\frac{1}{2} < h \le k + \tfrac{1}{2}\)  assuming the Riemann hypothesis.
